As the father, how do I get responsibility for my child if I’m not married or in a registered partnership?

As a father, you will only get parental responsibility if you first
acknowledge your child. After that you and the child’s mother can have it
recorded in the parental responsibility register that you wish to exercise joint
parental responsibility. The same applies if you have a cohabitation agreement
(samenlevingscontract) with the child’s mother.

The mother denies the father joint parental responsibility

If a child’s father wants to get joint responsibility but the mother does not
agree, the parents can’t submit a joint request for responsibility. But as the
father, you can still ask the court to assign joint responsibility. To do so you
will need a lawyer (advocaat). You must also have acknowledged the child.